Permutahedra and generalized associahedra

Hohlweg, Christophe; Langé, Christine; Thomas, Hugh

doi:10.1016/j.aim.2010.07.005

Cited by 99 publications

(172 citation statements)

References 27 publications

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“…This simplicial complex along with the generalized associahedron has become the object of intensive studies and generalizations in various contexts in mathematics, see, for instance, [12,27,40,48]. A generator s ∈ S is called initial or final in a Coxeter element c if (sc) < (c) or (cs) < (c), respectively.…”

Section: Cluster Complexesmentioning

confidence: 99%

Subword complexes, cluster complexes, and generalized multi-associahedra

2013

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In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k = 1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.

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Section: Cluster Complexesmentioning

confidence: 99%

Subword complexes, cluster complexes, and generalized multi-associahedra

2013

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“…The polytopality of the g-vector fan was studied by C. Hohlweg, C. Lange and H. Thomas [HLT11]. See also recent alternative proofs by S. Stella [Ste13] and V. Pilaud and C. Stump [PS15a].…”

Section: Theorem 50 ([Cfz02 Csz15]) the D-vector Fan (Or Compatibilmentioning

confidence: 99%

“…Since then, the associahedron has motivated a flourishing research trend with rich connections to combinatorics, geometry and algebra: polytopal constructions [Lod04, HL07, CSZ15, LP13], Tamari and Cambrian lattices [MHPS12,Rea04,Rea06], diameter and Hamiltonicity [STT88,Deh10,Pou14,HN99], geometric properties [BHLT09,HLR10,PS15b], combinatorial Hopf algebras [LR98,HNT05,Cha00,CP14], to cite a few. The associahedron was also generalized in several directions, in particular to secondary and fiber polytopes [GKZ08,BFS90], graph associahedra and nestohedra [CD06,Dev09,Pos09,FS05,Zel06,Pil13], pseudotriangulation polytopes [RSS03], cluster complexes and generalized associahedra [FZ03b,CFZ02,HLT11,Ste13,Hoh12], and brick polytopes [PS12,PS15a].…”

Section: Introductionmentioning

confidence: 99%

Compatibility fans for graphical nested complexes

Manneville¹,

Pilaud²

2017

Journal of Combinatorial Theory, Series A

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Abstract. Graph associahedra are natural generalizations of the classical associahedra. They provide polytopal realizations of the nested complex of a graph G, defined as the simplicial complex whose vertices are the tubes (i.e. connected induced subgraphs) of G and whose faces are the tubings (i.e. collections of pairwise nested or non-adjacent tubes) of G. The constructions of M. Carr and S. Devadoss, of A. Postnikov, and of A. Zelevinsky for graph associahedra are all based on the nested fan which coarsens the normal fan of the permutahedron. In view of the combinatorial and geometric variety of simplicial fan realizations of the classical associahedra, it is tempting to search for alternative fans realizing graphical nested complexes.Motivated by the analogy between finite type cluster complexes and graphical nested complexes, we transpose in this paper S. Fomin and A. Zelevinsky's construction of compatibility fans from the former to the latter setting. For this, we define a compatibility degree between two tubes of a graph G. Our main result asserts that the compatibility vectors of all tubes of G with respect to an arbitrary maximal tubing on G support a complete simplicial fan realizing the nested complex of G. In particular, when the graph G is reduced to a path, our compatibility degree lies in {−1, 0, 1} and we recover F. Santos' Catalan many simplicial fan realizations of the associahedron.keywords. Graph associahedra · finite type cluster algebras · compatibility degrees · compatibility fans.MSC classes. 52B11, 52B12, 05E45.

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“…We include, almost verbatim, the proof from [HLT11] for completeness. If k ∈ K p , the c-factorization of ws k is simply c K1 .…”

Section: C-sorting Björner and Wachsmentioning

confidence: 99%

“…We will need the following lemma to characterize reduced expressions for w 0 in S n coming from maximal length chains in T n . It is a very slightly modified version of a lemma from [HLT11]. If ws k = c K1 .…”

Section: C-sorting Björner and Wachsmentioning

confidence: 99%

Chains of maximum length in the Tamari lattice

Fishel

Nelson

2014

Proc. Amer. Math. Soc.

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Abstract. The Tamari lattice Tn was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, "The enumeration of such paths in Tamari lattices remains mysterious."The lengths of maximal chains vary over a great range. In this paper, we focus on the chains with maximum length in these lattices. We establish a bijection between the maximum length chains in the Tamari lattice and the set of standard shifted tableaux of staircase shape. We thus derive an explicit formula for the number of maximum length chains, using the Thrall formula for the number of shifted tableaux. We describe the relationship between chains of maximum length in the Tamari lattice and certain maximal chains in weak Bruhat order on the symmetric group, using standard Young tableaux. Additionally, recently, Bergeron and Préville-Ratelle introduced a generalized Tamari lattice. Some of the results mentioned above carry over to their generalized Tamari lattice.

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Permutahedra and generalized associahedra

Cited by 99 publications

References 27 publications

Subword complexes, cluster complexes, and generalized multi-associahedra

Subword complexes, cluster complexes, and generalized multi-associahedra

Compatibility fans for graphical nested complexes

Chains of maximum length in the Tamari lattice

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